Exact and Approximate Hierarchical Hub Labeling

Hub Labeling (HL) is a state-of-the-art technique for accelerating shortest path computation in road networks. By utilizing precomputed node labels, it can answer distance queries in microseconds on continent-sized networks. The optimization goal is to get correct query results with a minimum number of labels. There is an $\mathcal{O}(\log n)$ approximation algorithm for the size of an HL with a running time of $\mathcal{O}(n^3\log n)$. However, existing practical implementations rely mostly on heuristics for a special type of HL, so called Hierarchical HL (HHL). Deciding whether a graph admits a labeling of size at most $k$ is NP-hard for both HL and HHL. For HHL, an $\mathcal{O}(\sqrt n \log n)$ approximation algorithm (called w-HHL) is known. In this article, we devise an exact HHL algorithm for general graphs. We also show that the exact algorithm transfers to Hierarchical Landmark HL (HLHL), which is a generalization of HHL. Moreover, we prove that w-HHL provides a constant factor approximation on trees and investigate for the first time the practical performance of HHL approximation algorithms. We also compare the resulting label sizes to heuristic HHL and HLHL. Our experimental results offer novel insights and show that commonly used methods for HHL are noticeably outperformed by w-H(L)HL on general graphs as well as trees.